3. TARGET THEORY Much of the experimental work in radiation biology involves dose-response
relationships. The fraction of damaged biological entities C*/C Figure 12. Dose-response curves. (a) zero-order process, (b) first-order process, (c) second-order process, (d) four-hit process. (Adapted from Grossweiner.) phenomenologicalapproach to biological radiation damage which provides insights into the damage mechanisms in terms of the survival curves. The basic assumption of target theory is that an observable response takes place when a localized region of a Figure 13. Survival curves for inactivation of E. coli by biological system or a target absorbs one or more units of energy referred
to as hits. The hit concept implies that radiation damage is a discrete process. Every
damaging event may not constitute a hit owing to the location of the lesion and the action
of repair processes. Target theory assumes that radiation damage is a random process. The
distribution of hits should follow a (6): p(n) = Exp(-n Thus, if the average number of errors per chapter is 5, the probability
that any one chapter has exactly one error is: 5 x Exp(-5)/1! = 0.0337 and the probability
of exactly five errors is: 5 (7): For direct action n (8): n where (9): 1/ Thus, 1/ (10): MW where W is in eV and Many survival curves have an initial curved region followed by a steeper
linear decrease at high dose. The initial stage is attributed to sub-lethal damage. This
response can occur if more than one target must be damaged or if more than one hit on a
single target is required to initiate the observed effect. Alternatively, the earlier
damage may undergo (11): Figure 14 shows semi-logarithmic plot of
Equation (11) for different values of n. The low-dose shoulder corresponds to sub-lethal
damage during the initial period
while hits are accumulating. Expanding the right side of Equation (11) shows that in the
limit of high dose: Figure 14. Theoretical survival curves for n hits on a single target. The extrapolation of the high dose line gives the hit number. (Apadpted from Zimmer.)
D_{o} + D_{th}.
D_{37} is higher than in the single-hit model because all m targets must
accumulate at least one hit before 63% of the systems respond. D_{o} is the
equivalent of the one-hit D_{37} as defined by Equation (5). More
complicated models have been proposed for radiation inactivation of mammalian cells. The
assumption that a cell has g equivalent sites, each of which is an r-target entity, and
that damage to any site is lethal leads to:(12): The extrapolation number for this case is r (13): was proposed for mammalian cell damage in the theory of
3.1. Effect of LET on Radiation Sensitivity Comparisons between
4. INDIRECT ACTION OF IONIZING RADIATION Indirect action in aqueous and biological systems involves the radiation
chemistry of liquid water. The primary products of water radiolysis are generated with a
non-uniform spatial distribution owing to the inhomogeneous distribution of the initial
energy loss. The analysis of the early-time regime is based on spur theory. Those primary
species that survive spur reactions eventually attain a random spatial distribution in the
medium. The later-time regime obeys conventional
3.1. Homogeneous Reaction Kinetics The kinetics of chemical reactions are described in terms of Conventional reaction kinetics express the reaction rate in the form of
total differential equations which are then solved for the relevant concentrations. A
uniform spatial distribution of the constituents is assumed, which rules out the
application of this approach to spur reactions. A unimolecular A v products If the reaction rate does not depend on the amount of species-A the integrated rate equation is: (14): [A] - [A] where [A] is the molar concentration of A, [A] (15): Log where k A + A v products with the rate equation: (16): [A] = [A] where k A + B v The general solution is easily found for different starting amounts of A
and B. A special case obtains if [B] >> [A]. The reaction is then of pseudo-first
order and follows Equation (15) with k
3.2 Diffusion-Limited Rate Constants Many bimolecular reactions of small free radicals are exothermic and
generate energetic or "hot" products in the gas phase. When the reaction takes
place in solution, the excess kinetic energy is carried away by nearby solvent molecules -
this phenomenon is the (17): k where R (18): a = kT/3phD where D is the diameter of the molecule and h is the viscosity of the medium. (The
viscosity of water is 1 poise at 20.2EC; 10 poise = 1 J-s/m Various techniques have been devised to measure the rate of fast chemical
reactions. In Figure 15. Schematic of pulse radiolysis apparatus for kinetic spectrophotometry. This system follows the optical absorption of the sample at single wavelengths before, during, and after a pulse of ionising radiation. The spectra of short-lived reaction products are analyzed by plotting the optical signals at different wavelengths and fixed time delays after the irradiation pulse. (Adapted from Swallow.) Pulse radiolysis studies have led to the identification of many
short-lived intermediates and measurements of reaction rate constants. The
3.3. Steady-State Competition Kinetics Steady-state reaction kinetics are based on the assumption that the short-lived intermediates are present in very low concentrations under continuous irradiation. Consider the indirect action of ionising radiation on biological entities (T) irradiated in the presence of an additive (A) which competes for the short-lived radical intermediates (R). The relevant reactions are: H R + A v products (k R + T vT’ (k where Y denotes a primary radiolysis reaction, k (19): 1/ where G
3.4.
Conventional bimolecular reaction kinetics are inaccurate for reactions of
small entities with widely-separated large targets owing to depletion of the reactive
entity near the surface of a large target faster than can be replenished by diffusion.
This section was written by the author to exemplify a practical theoretical approach. The
reaction volume ( (20): n where Q is the average number of small entities per unit volume of medium
and c (21): n where (22): where d (23): where L (24): d where k (25): n For a one-hit case (26): Equation (26) gives the radiation sensitivity for reactions of a reactive
radical with a large spherical target. In the limit of a long-lived radical compared to
the diffusion length (d (27): The term in (..) is the same as diffusion-limited rate constant for
bimolecular reactions of an uncharged radical; see Equation (17). Equation (27) is the
same result as calculated by homogeneous reaction kinetics in which reactions of radicals
with targets compete with scavenging. In the other limit of a very large target (R
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